A Tits alternative for endomorphisms of the projective line

We prove an analog of the Tits alternative for endomorphisms of P ¹. In particular, we show that if S is a finitely generated semigroup of endomorphisms of P ¹ over C, then either S has polynomially bounded growth or S contains a nonabelian free semigroup. We also show that if f and g are polarizable maps over any field of any characteristic and Prep.f / 6D Prep.g/, then for all sufficiently large j, the semigroup hf ^j ; g^j i is a free semigroup on two generators.